The representative of the quantum algorithm are Shor' factoring[1]} and Grover's search[2]} algorithms.
Few years ago another quantum algorithm[3]} called
Harrow-Hassidim-Lloyd (HHL) algorithm was developed. It is a quantum algorithm, which can compute the inverse of sparse matrix.
If \(A\)  is a \(N \times N\)  sparse matrix, HHL algorithm completes the inversion task with a runtime of \({\cal O} \left( s^2 \kappa ^2 \log (N) / \epsilon \right)\) , where
\(s\)  is the maximum number of non-zero entries in row or column, \(\kappa \)  the condition number, and \(\epsilon \)  the precision.
Since same task can be implemented in the classical computer with a runtime of \({\cal O} \left( s \kappa N \log (1 / \epsilon ) \right)\)  even though the most efficient algorithm is adopted,
one can say that the HHL algorithm improves exponentially in the matrix inversion task over the best classical algorithm.
This algorithm is based on the efficient Hamiltonian simulation[4]}. Subsequently, there was a proposals to deal with dense matrices[5]}. Also a hybrid algorithm[6]} was proposed, where
both classical and quantum computers are used appropriately.
